Optimal fault-tolerant Jacobian matrix generators for redundant manipulators

Abstract

Abstract-The design of locally optimal fault-tolerant manipulators has been previously addressed via adding constraints on the bases of a desired null space to the design constraints of the manipulators. Then by algebraic or numeric solution of the design equations, the optimal Jacobian matrix is obtained. In this study, an optimal fault-tolerant Jacobian matrix generator is introduced from geometric properties instead of the null space properties. The proposed generator provides equally fault-tolerant Jacobian matrices in R 3 that are optimally faulttolerant for one or two locked joint failures. It is shown that the proposed optimal Jacobian matrices are directly obtained via regular pyramids. The geometric approach and zonotopes are used as a novel tool for determining relative manipulability in the context of fault-tolerant robotics and for bringing geometric insight into the design of optimal fault-tolerant manipulators.

A probabilistic approach for measuring the fault tolerance of robotic manipulators

“…The SVD of the faulty manipulator with the k-th joint locked is denoted by (5) where U E R m x m and V E R(I1-l )x(l1-l ) are orthogonal matrices and L E R m x(l1-l ) is a diagonal matrix. For ( kal +k a2 + ... +k a m -l +k a m) (7) m Geometric mean of the singular values (related to manipulability) :…”

“…The development of fault tolerant manipulators has been possible through the use of kinematic redundancy [2], [5], [13], [14], [19], [18], [22] along with applying fault-tolerant control strategies [7], [11], [13], [16], [15], [20], [21], [23]. The optimal kinematic structures and fault tolerant controllers for the manipulators are commonly obtained by the optimization of the fault tolerance indices [2], [5], [19], [18], [22].…”

“…There are different fault tolerance indices that have been used for optimization purposes, such as worst-case relative manipulability [18], [22], [5], worst-case minimum singular value [13], [14], and worst-case condition number [3]. These measures are calculated locally and are useful for tasks that do not require large motions.…”

Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions

“…, r. This function has extensive applications in mathematical physics, for example, in [14] it was used to model post-buckling of beams. While T (x) is one of the fundamental functions in engineering sciences, which arises broadly in regions including plasticity theory [30], nonsmooth variational problems [11], structural optimization problems [3], information theory [8], network communication systems [7,4,9], and robot manipulator designing [1,2,28]. In numerical analysis, the function T (x) is often used to deal with minimax problems [25,26,27,29].…”

Reliability maps for probabilistic guarantees of task motion for robotic manipulators

“…fault analysis and fault-tolerant motion planning or faulttolerant controllers) or the design of the manipulators (e.g. fault-tolerant serial 15 manipulator or parallel manipulators [10][11][12][13][14]). Within the literature of the faulttolerant control of manipulators, various subjects such as failure analysis, fault detection, fault isolation, fault identification, and fault accommodation of the manipulators are studied [15][16][17].…”